Second-Order Josephson Effect

• Second-order Josephson effect is a phenomenon characterized by a current-phase relation dominated by a sin2φ harmonic, leading to π periodic behavior.
• It arises from diverse microscopic mechanisms such as spin filtering, diffusive SIS corrections, and coherent Andreev processes in high-transparency junctions.
• Experimental signatures include half-integer Shapiro steps, halved magnetic interference patterns, and AC manifestations, influencing superconducting circuits and diode functionalities.

Searching arXiv for recent and foundational papers on the second-order Josephson effect. The second-order Josephson effect denotes a current–phase relation (CPR) in which the leading superconducting supercurrent varies as $\sin 2\phi$ rather than $\sin \phi$, or in which a substantial $\sin 2\phi$ harmonic coexists with the conventional first harmonic. In Fourier form, the CPR may be written as $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$, with the second-order effect corresponding either to a dominant or pure $I_2$ term, or to physical phenomena specifically associated with the second harmonic (Pal et al., 2014, Osin et al., 2021). Across contemporary literature, this effect appears in several distinct settings: spin-filter Josephson junctions with purely second-harmonic transport, diffusive SIS tunnel junctions where $I_2$ emerges as a self-consistent correction, high-transparency semiconductor and topological junctions where large $I_2/I_1$ is measured, voltage-biased junctions where Higgs-mode dynamics enhance the $2\omega_J$ AC component, multicomponent superconductors with second-order intercomponent couplings, and excitonic insulators where the effective Josephson coupling is intrinsically $\cos 2\theta$ rather than $\cos\theta$ (Pal et al., 2014, Zhang et al., 2022, Sivakumar et al., 2024, Lahiri et al., 2024, Peng et al., 27 May 2026, Sun et al., 2021).

1. Definition and current–phase structure

In the conventional Josephson effect, the supercurrent through a weak link is described by

$\sin \phi$0

where $\sin \phi$1 is the gauge-invariant phase difference and $\sin \phi$2 is the first-harmonic critical current (Pal et al., 2014). More generally,

$\sin \phi$3

with higher harmonics becoming relevant in the presence of transmission resonances, multiple Andreev reflections, or magnetic and spin-active interfaces (Pal et al., 2014).

The term “second-order Josephson effect” is used in two closely related senses in the literature. In one sense, it refers to a CPR whose dominant nonlinear contribution is the second harmonic $\sin \phi$4, as in spin-filter, semiconductor, and topological Josephson junctions (Pal et al., 2014, Zhang et al., 2022, Sivakumar et al., 2024). In another sense, it refers to an underlying coupling energy proportional to $\sin \phi$5 or $\sin \phi$6, implying a $\sin \phi$7-periodic phase dependence and two degenerate minima at phase differences differing by $\sin \phi$8, as in excitonic insulators and in second-order intercomponent couplings of multicomponent superconductors (Sun et al., 2021, Peng et al., 27 May 2026).

A central consequence is doubled phase periodicity. A junction with $\sin \phi$9 has period $\sin 2\phi$0 rather than $\sin 2\phi$1 (Pal et al., 2014). In AC transport, the second harmonic produces a supercurrent oscillating at $\sin 2\phi$2 when the condensate phase evolves as $\sin 2\phi$3 under voltage bias (Lahiri et al., 2024). In interference phenomena, a dominant $\sin 2\phi$4 term leads to half-periodic SQUID oscillations and halved Fraunhofer-like magnetic periods (Pal et al., 2014, Zhang et al., 2022).

2. Microscopic origins in superconducting junctions

One microscopic route to a second-order CPR is suppression of the singlet first harmonic by spin filtering. In NbN/GdN/NbN mesa devices, GdN acts as a spin-dependent tunnelling barrier with polarization $\sin 2\phi$5 and up to approximately $\sin 2\phi$6 at $\sin 2\phi$7 (Pal et al., 2014). As $\sin 2\phi$8, singlet Cooper-pair tunnelling requires both spin channels and the first-harmonic contribution $\sin 2\phi$9 (Pal et al., 2014). If spin mixing at the superconducting interfaces converts singlets into equal-spin triplets, coherent transport of two triplet pairs yields directly

$I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$0

with no first-harmonic term (Pal et al., 2014). In the Green’s-function formulation quoted there,

$I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$1

where the square denotes coherent transfer of two triplet pairs, and the amplitude $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$2 scales as $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$3 (Pal et al., 2014).

A different microscopic regime is the diffusive SIS tunnel junction. In a planar SIS geometry with diffusive superconducting banks, a fully self-consistent perturbation theory in the small parameter $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$4 yields

$I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$5

with

$I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$6

(Osin et al., 2021). In that formulation, the second harmonic is a small positive correction whose magnitude is explicitly temperature dependent. The same treatment shows that at second order the spectral phase $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$7 departs from the condensate phase $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$8, i.e. $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$9 (Osin et al., 2021).

High transparency provides another established route. In a short single-mode ballistic SNS junction with transparency $I_2$0, the Andreev bound-state spectrum

$I_2$1

yields a nonsinusoidal CPR whose harmonic expansion contains a sizable $I_2$2 term (Zhang et al., 2022). In planar Al/InAs-quantum-well junctions, a two-component model with $I_2$3 fits multiple measurements, and analysis including loop inductance suggests that the sign of the second harmonic is negative (Zhang et al., 2022). In 1T-PtTe$I_2$4, the second harmonic is attributed to topological spin-momentum-locked states that promote coherent Andreev processes, with the relative phase between the $I_2$5 and $I_2$6 components tunable by magnetic field (Sivakumar et al., 2024).

The sign of $I_2$7 is not merely a fitting detail. In altermagnetic Josephson junctions, the truncated CPR

$I_2$8

can be forward- or backward-skewed depending on the sign of $I_2$9 (Sun et al., 2024). For $I_2$0, the fit gives $I_2$1 with a minus sign, while for $I_2$2, $I_2$3 with a plus sign (Sun et al., 2024). This directly links second-harmonic physics to skewness, $I_2$4-junction behavior, and field-tuned $I_2$5–$I_2$6 transitions.

3. Experimental signatures and diagnostic criteria

The most direct experimental signatures of a second-order Josephson effect are half-periodicity in magnetic interference, half-integer Shapiro steps, and explicit Fourier extraction of the $I_2$7 component. In spin-filter NbN/GdN/NbN junctions, devices with GdN thickness at or above $I_2$8 show a measured interference period

$I_2$9

and the ratio of second-lobe width $I_2/I_1$0 to first-lobe width $I_2/I_1$1 converges to unity once the barrier’s internal flux is accounted for, matching the prediction for $I_2/I_1$2 (Pal et al., 2014). No evidence of $I_2/I_1$3 contributions is reported in those devices (Pal et al., 2014).

In planar superconductor–semiconductor junctions, the second harmonic is identified through three complementary probes (Zhang et al., 2022). First, dc-SQUIDs exhibit half-periodic oscillations tunable by gate voltages and magnetic flux. Second, single-junction diffraction patterns show kinks near half-flux quantum. Third, microwave irradiation produces half-integer Shapiro steps. In the cited Al/InAs devices, the best fit in a symmetric SQUID configuration gives $I_2/I_1$4 for both junctions, while similar signatures are also observed in Sn/InAs devices (Zhang et al., 2022).

In the Ta$I_2/I_1$5Pd$I_2/I_1$6Te$I_2/I_1$7 asymmetric edge interferometer, half-integer Shapiro steps appear at voltages

$I_2/I_1$8

with $I_2/I_1$9, directly signaling a non-zero $2\omega_J$0 term (Li et al., 2023). Those steps persist over a broad range of microwave drive powers (Li et al., 2023). The same work reports antisymmetric second-harmonic transport in lock-in measurements, where

$2\omega_J$1

and $2\omega_J$2 is antisymmetric about $2\omega_J$3, closely tracking the diode-current asymmetry $2\omega_J$4 (Li et al., 2023).

In 1T-PtTe$2\omega_J$5, the CPR is extracted from Fraunhofer critical-current oscillations under $2\omega_J$6 and Fourier-transform methods. The data exhibit both a fundamental $2\omega_J$7 period from $2\omega_J$8 and a strong $2\omega_J$9 component from $\cos 2\theta$0 (Sivakumar et al., 2024). In one junction, $\cos 2\theta$1 at $\cos 2\theta$2 implies $\cos 2\theta$3 and $\cos 2\theta$4 (Sivakumar et al., 2024).

These signatures are not interchangeable in evidentiary strength. One source explicitly describes the magnetic interference pattern $\cos 2\theta$5 as the “most unambiguous probe of the CPR” (Pal et al., 2014), while another notes that half-integer Shapiro steps can also arise from vortex phase locking or non-equilibrium quasiparticles and should therefore be treated as corroborating rather than primary evidence (Zhang et al., 2022). This suggests that robust identification of a second-order Josephson effect is strongest when multiple probes concur.

4. Dynamical and AC manifestations

Under DC voltage bias, the phase evolves as

$\cos 2\theta$6

and a second-harmonic contribution produces current oscillations at $\cos 2\theta$7 (Lahiri et al., 2024). In transparent junctions between single-band $\cos 2\theta$8-wave superconductors, allowing for order-parameter amplitude dynamics modifies the Josephson current to

$\cos 2\theta$9

with $\cos\theta$0 (Lahiri et al., 2024). If $\cos\theta$1, then

$\cos\theta$2

where

$\cos\theta$3

(Lahiri et al., 2024).

The enhancement mechanism is resonant excitation of the superconducting Higgs or amplitude mode. In the effective Lagrangian,

$\cos\theta$4

with $\cos\theta$5 (Lahiri et al., 2024). The Josephson term acts as a periodic drive, and the response is filtered by the retarded Higgs susceptibility $\cos\theta$6, which peaks at $\cos\theta$7 (Lahiri et al., 2024). Near resonance, the $\cos\theta$8 component can exceed the $\cos\theta$9 component when the equilibrium gaps are sufficiently asymmetric and the transparency is high (Lahiri et al., 2024).

The same paper gives a numerical illustration in a two-dimensional model with $\sin \phi$00, $\sin \phi$01, $\sin \phi$02, $\sin \phi$03, bandwidth $\sin \phi$04, transparency $\sin \phi$05 and damping $\sin \phi$06, where the $\sin \phi$07 current eventually exceeds the $\sin \phi$08 component when $\sin \phi$09 (Lahiri et al., 2024). This is a dynamical second-harmonic dominance rather than a static pure $\sin \phi$10 CPR.

AC manifestations also arise in non-Higgs contexts. In diffusive SIS junctions, the small $\sin \phi$11 term shifts the phase of maximum critical current away from $\sin \phi$12 and can produce half-harmonic Shapiro steps (Osin et al., 2021). In excitonic insulators, by contrast, the phase equation under strong drive gives $\sin \phi$13, and because the current is $\sin \phi$14, the AC Josephson frequency remains $\sin \phi$15 despite the $\sin \phi$16-periodic phase dependence (Sun et al., 2021).

5. Alternative formulations beyond conventional superconducting weak links

The second-order Josephson concept extends beyond ordinary Cooper-pair tunnelling through a weak link. In electron–hole bilayers with excitonic order formed by orbitals of opposite parity, integrating out fermions to second order in the interlayer tunnelling yields an effective phase Lagrangian

$\sin \phi$17

(Sun et al., 2021). The free energy therefore contains $\sin \phi$18, and the interlayer current obeys

$\sin \phi$19

(Sun et al., 2021). The minima at $\sin \phi$20 are degenerate, and a voltage pulse with $\sin \phi$21 switches between them (Sun et al., 2021). In Ta$\sin \phi$22NiSe$\sin \phi$23, reflection-symmetry analysis is argued to place the system in precisely this class (Sun et al., 2021).

In multicomponent superconductivity, second-order Josephson couplings occur between condensates rather than across a spatial junction. In the three-component Ginzburg–Landau model compatible with the 3Q pair-density-wave state, the free-energy density includes terms

$\sin \phi$24

with no conventional $\sin \phi$25 coupling (Peng et al., 27 May 2026). These couplings are invariant under both time reversal and $\sin \phi$26-phase flip, but in frustrated regimes the ground state can spontaneously break both symmetries (Peng et al., 27 May 2026). The theory identifies five distinct ground states: an 8-fold degenerate frustrated state and four 4-fold degenerate non-frustrated phase-locked states, and numerical analysis finds a Higgs–Leggett mode unique to the frustrated region (Peng et al., 27 May 2026). This is a second-order Josephson effect in the intercomponent phase sector.

A related but distinct framework is the bilayer XY model with second-order Josephson coupling

$\sin \phi$27

(How et al., 25 Jul 2025). The term $\sin \phi$28 pins the interlayer phase difference to $\sin \phi$29 or $\sin \phi$30, leaving a common $\sin \phi$31 and a discrete $\sin \phi$32 symmetry (How et al., 25 Jul 2025). In the dual formulation, the second-order Josephson term maps to a two-dimensional noncompact $\sin \phi$33 gauge field, and the resulting theory predicts that the only transition out of the low-temperature ordered phase is an Ising transition driven by condensation of $\sin \phi$34 domain-wall loops (How et al., 25 Jul 2025). The paper argues that point-defect-based Coulomb-gas methods miss the relevant excitations in this regime (How et al., 25 Jul 2025).

6. Device consequences, nonreciprocity, and circuit applications

A large or dominant second harmonic strongly alters device functionality. In the Ta$\sin \phi$35Pd$\sin \phi$36Te$\sin \phi$37 edge interferometer, the generalized interferometric CPR includes both first and second harmonics on each branch,

$\sin \phi$38

and the second-harmonic term is an important element in generating a Josephson diode effect (Li et al., 2023). The reported diode efficiency reaches approximately $\sin \phi$39 in one device and up to approximately $\sin \phi$40 in another at $\sin \phi$41, with switching powers of approximately $\sin \phi$42 and approximately $\sin \phi$43, respectively (Li et al., 2023).

In 1T-PtTe$\sin \phi$44, the second harmonic is directly correlated with a large intrinsic Josephson diode effect, and the relative phase between the $\sin \phi$45 and $\sin \phi$46 harmonics is tunable with in-plane magnetic field (Sivakumar et al., 2024). The free-energy expansion quoted there,

$\sin \phi$47

implies a field-tunable phase shift of the second-harmonic component (Sivakumar et al., 2024). The diode efficiency reaches approximately $\sin \phi$48 at $\sin \phi$49 (Sivakumar et al., 2024).

In altermagnetic junctions, electric and Zeeman fields tune not only the magnitude of $\sin \phi$50 but also its sign and the skewness of the CPR (Sun et al., 2024). The ratio $\sin \phi$51 crosses unity near $\sin \phi$52, where the junction becomes a $\sin \phi$53-junction, and can be tuned by gate potential $\sin \phi$54 or in-plane field $\sin \phi$55 (Sun et al., 2024). The same work reports field-induced $\sin \phi$56–$\sin \phi$57 transitions and a regime where the critical current increases with Zeeman field, which it describes as surprising because supercurrents are typically suppressed by magnetic fields (Sun et al., 2024).

The second harmonic also affects superconducting circuits more broadly. In a Josephson traveling-wave parametric amplifier with generalized CPR

$\sin \phi$58

the Josephson potential becomes

$\sin \phi$59

(Guarcello et al., 2 Feb 2025). Numerical simulations for an array of $\sin \phi$60 cells with $\sin \phi$61, $\sin \phi$62, $\sin \phi$63, $\sin \phi$64, pump frequency $\sin \phi$65 and signal frequency $\sin \phi$66 show that the weighting of the second harmonic changes the gain profile, phase-space structure, and stability; the maximum gain reaches approximately $\sin \phi$67 without dispersion engineering, peaking near $\sin \phi$68 (Guarcello et al., 2 Feb 2025). A plausible implication is that second-harmonic engineering is relevant not only to equilibrium CPR physics but also to nonlinear microwave design.

7. Conceptual distinctions, limitations, and open issues

The second-order Josephson effect should not be conflated with a single universal microscopic mechanism. In spin-filter junctions, the effect is linked to suppression of singlet tunnelling and coherent transfer of two triplet pairs (Pal et al., 2014). In diffusive SIS junctions, it is a perturbative self-consistent correction to conventional tunnelling theory (Osin et al., 2021). In high-transparency semiconductor and topological junctions, it is associated with Andreev-bound-state structure and coherent higher-order transport (Zhang et al., 2022, Sivakumar et al., 2024). In Higgs-driven AC transport, it arises from nonequilibrium order-parameter dynamics (Lahiri et al., 2024). In excitonic insulators and multicomponent superconductors, it is built into the effective phase energy as a fundamental $\sin \phi$69 coupling (Sun et al., 2021, Peng et al., 27 May 2026).

A recurring point of contention concerns interpretation of the second harmonic as coherent $\sin \phi$70 transport. One source explicitly defines the $\sin \phi$71 term as a distinct four-electron process in 1T-PtTe$\sin \phi$72 (Sivakumar et al., 2024), while another frames large $\sin \phi$73 in planar semiconductor junctions more cautiously, stating that the microscopic origins remain to be understood and that alternative explanations can account for some but not all evidence (Zhang et al., 2022). This suggests that the language of “quartets” or “$\sin \phi$74 transfer” is well motivated in some settings but not yet universally established across all large-$\sin \phi$75 experiments.

Another important distinction concerns robustness. Standard theory predicts higher harmonics to be extremely sensitive to barrier thickness and temperature, especially near $\sin \phi$76–$\sin \phi$77 transitions. The spin-filter experiments report instead a purely second-harmonic CPR that is insensitive to barrier thickness beyond approximately $\sin \phi$78 and persists from $\sin \phi$79 to at least $\sin \phi$80, which is taken to imply that standard spin-inactive tunnelling theory is not applicable to spin-dependent barriers (Pal et al., 2014). By contrast, in self-consistent SIS theory the second harmonic is small and parametrically controlled by $\sin \phi$81 or $\sin \phi$82 (Osin et al., 2021).

The broader significance of the second-order Josephson effect lies in its combination of doubled periodicity, symmetry-selective coupling, and nonlinear tunability. The literature connects it to intrinsically protected $\sin \phi$83–$\sin \phi$84 qubits, built-in phase bias elements, rapid single-flux-quantum logic, protected phase slips, superconducting diodes, low-power memory, higher-order coherent transport, and spectroscopic access to collective modes (Pal et al., 2014, Li et al., 2023, Sivakumar et al., 2024, Sun et al., 2021). A plausible implication is that the second harmonic has moved from being a small correction in Josephson phenomenology to a design principle spanning superconducting weak links, multicomponent condensates, and correlated electron-hole systems.